After having critically assessed the general motivations to introduce quantum logic as a consequence of the physical content of QM, in this section we present our main argument according to which it is possible to assign a distributive lattice to quantum propositions. Remarkably, in what follows we will see that the non-distributivity of the quantum logic lattice originates neither from non-commuting observables nor from superpositions of quantum mechanics: it is a property of the lattice of subspaces of any vector space, and of the lattice of subgroups of any group more in general (cf. [1]}). In this regard, indeed, we also show that taking into account the space of classical probability distributions one can apply a non-distributive lattice to classical propositions. This fact is due to the similar structures of the state space of quantum mechanics and the space of classical probability distributions.
